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DOCUMENT RESUMETM 003 003ED 079 400AUTHORTITLEINSTITUTIONREPORT NOPUB DATENOTEEDRS PRICEDESCRIPTORSStroud*, T. W. F.Combining Unbiased Estimates of a Parameter Rnown tobe Positive.Educational Testing Service, Princeton, N.J.ETS-RB-73-47Jun 7328p.AF-f 0.65 HC- 3.29Calculation; Mathematical Applications; *MathematicalModels; Measurement Techniques; *StatisticalAnalysis; *Statistical Bias; StatisticsABSTRACTThe statistician has n independent estimates of aparameter he knows is positive, but, as is the case incomponents-of-variance problems, some of the estimates may benegative. If the n estimates are to be combined into a single number,we compare the obvious rule, that of averaging the n values andtaking the positive part of the result,- with that of averaging thepositive parts. Although the estimator generated by the second ruleis not consistent, it is shown by numetical calculation that forsmall n it has a smaller mean square error than the first over aconsiderable region of the parameter space, and that for n 2 or 3the second is minimax relative to the firSt over a region consistingof almost the whOle parameter space. The distribution of each of then- estimates is assumed to be either GausSian or the distribution of aweighted difference of two independent chi-squares with known degreesof freedom, as in one way components of variance. Some other simplycalculated estimators, including the positive part of the median, arestudied for the chi-square difference case with (2,2)- a degrees offreedom and n 3. (Author)

FILMED FROM BEST AVAILABLE COPYU S.DEPARTMENT Of HEALTH,EDUCATION WIELCADENATIONAL INSTITUTE OFEDUCATIONRB-73-47THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIGINATING IT POINTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRESENT OFFICIAL NATIONAL INSTITUTE OFEDUCATION POSITION OR POLICYACOMBINING UNBIASED ESTIMATES OF A PARAMETERKNOWN TO BE POSITIVERcT. W. F. StroudcCTQueen's UniversityandEducational Testing ServiceNev4t0This Bulletin is a draft for interoffice- circulation.Corrections and suggestions for revision are solicited.The Bulletin should not be cited as a reference withoutthe specific permission of the author.It is automati-cally superseded upon formal publication of the material.Educational Testing ServicePrinceton, New JerseyJune 1973

COMBINING UNBIASED ESTIMATES OF A PARAMETER KNOWN TO BF POSITIVET. W. F. StromeABSTRACTThe statistician hasnindependent estimates of a parameter he knowsis positive but, as is the case in components -of- variance problems, some ofthe estimates may be negative.estimates are to be combined intonIf thea single munber, we compare the obvious rule, that'of averaging thenvalues and taking the positive part of the result, with that of averagingthe positive parts.Although the estimator generated by the second rule isnot consistent, it is shown by numerical calculation that for smallnithas a smaller mean-souare error than the first over a considerable regionof the parameter space, and that forn 2 or 3the second is minimalrelatiVe to the first over aregion consisting of almost the whole parameterspace.The distribution of each of thenestimates is assumed to beeither Gaussian or the distribution of a weighted difference of two independent chi-squares with known degrees of freedom, as in one-way components ofvariance.Some other simply calculated estimators, including the positivepart of the median, are studied for the chi-square difference case with-(2,2) degrees of freedom andn 3 .

COMBINING UNBIASED ESTIMATES OF A PARAMETER KNOWN TO BE POSITIVE1.INTRODUCTIONSometimes, most notably in components-of-variance problems, a statistician is trying to estimate a parameter he knows is positive, but in usingstandard-unbiased estimation techniques he obtains an estimate which isnegative.When this happens, the statistician would usually estimate theparameter by zero, since in so doing he is coming closer to the true parameter value than the original estimate.Occasionally the statistician may have several unbiased estimates ofthe same parameter arising from independent sources.The statistician willwant to combine the raw data sets and treat them as one data set; however,this may not always be possible.There may be nuisance parameters whichvary from one source to another.Or the computational procedure mayrequire too much memory to accommodate all the data at once.This mayhappen in using Rao's MINQUE technique [8], where matrices the size of thedata set are handled, or Henderson's third method [4,9] with a large numberof groups.(For a problem where either or both of these techniques arecalled for, see [11]).Finally, the statistician may not be able to com-bine the pbservations due to not being supplied with the raw data.let us assume that from each source the statistician has nothingmore than an unbiased estimate (which can be positive or negative) of acommon unknown parameter value which is necessarily positive.We studyalternative procedures for combining the estimates into a single number.The estimates are assumed to be identically distributed.(This does notcover the situation of-nuisance parameters varying over experiments, but,

-2-when caution is used, some features of the results reported here may stillapply.)The main part of this paper is devoted to studying two simple techniquesfor combining the estimates.A few other-procedures are treated in alimited way in the last section.We study the performances of the twotechniques when the distribution of the estimates is normal and when it isthe weighted difference of two independent chi-squares with known degreesof freedom.,4 The latter model is the correct one for one-way components ofvariance, and may be considered as a prototype for more complicated problems.As the degrees of freedom vary, a range of distributional shapes is Obtained.The normal distribution is, of course, the limiting case as bothdegrees of freedom become large.THE TWO-ESTIMATORS-AND-THEIR MEAN SQUARE ERRORS2.LetXXI'2be a set of independent random variables, each. Xnmember representing the estimate of-the unknown parameteru based onare assumed to be identically distributed withXione source.Theexpectationu E(Xi)the standard deviationand one nuisance parameter which is taken to bea a(Xi) .equal to zero, although anyXiWe know thatuis greater than orhas a positive probability of beingnegative.Denote by XIwhich equalsn 1 ,X,Xithe positive part ofwhenXi 0X.,i.e., the random variableand which equals zero otherwise.is the obvious estimator ofµ .WhenWhenn 1 , one can

-3--either average thevalues and then take the positive part or take theXipositive parts before averaging.the first method, bynormality.a,Denote,(R) ,the estimator obtained bysince it is the maximum-likelihood estimator under(E4)/n byWe denote the second estimatorµ , since it is theThese two estimators were con-arithmetic mean of a set of quantities.sidered by,Sirotnik [10] in a mental testing application.We note thattoE(XI)µis not consistent, since it converges in probabilityP(Xi 0)E(Xil Xi 0) P(Xi 0)E(Xiki even moderate.n ,E(XI) 0) P(Xi a)E(XilXi 0) E(Xi)has better properties and isaOne's first reaction to this fact is thatthus the preferred estimator.i.e.,nwhich is always greater thanThis is certainly true whennis large, orThe reason for presenting this article is that, for smalln performs better over a reasonably large portion of the parameterspace, assuming as we do that theXiare either Gaussian or the weighteddifference of 'two independent chi-squares.The comparison of the estimators' performances is based on mean squareerror (MSE), which is defined as the expected squared difference betweenthe parameter and its finally estimated value, and which equals the variance plus the square of the bias.It will be seen that formulas for theMSE contain a-proportionality factor of2a ; for this reason the compari-sons to be presented are in terms of MSE/a2.insert Table 1 about here

-4Table 1 is a summary of the relative performance of2measured by MSE/aWhen.pia 1/2such that2The value of MSE/aexceeds 1/2.for bothaandnapproaches its theoretical upper boundgiastrict consideration to the region2MSE/aofµ,asg/a K,and, depending on tha distribution of1/nHowever, if we re-M .K M , the maximumfor anyover this region is. less than that ofand eventuallyincreases tog1asais generally better, butis increasing ina11' is as good as11eandincreases, and the critical value ofthe advantage becomes minimal as npia,µan 2 or 3whenxi , sometimes also whennor 5 .The relative advantage of11overwhenag/a 1/2also dependson thedistributionoftheL.In the next section some numericalresults are presented for the normal case and for the chi-square difference with certain specified degrees of freedom.shall see in detail under what conditions3.When 'nthis caseais clearly a poor competitor towill be close tog .is the preferred estimator.NUMERICAL CALCULATIONS AND GRAPESis large,to be larger thangFrom these results wegandµg ,siace inwill be close to a number knownLet us therefore see what happens fornrangingfrom2to5,andtostartwithassumetheX.to be normally distributed.The calculations are obtained from formulas (4.3) and (4.4).We note that

-5-for,bothanda1-1the MSE equalsa2times a function of the standard-ized mean m f p/a tthe reciprocal of the coefficient of variation).For this reason we graph MSE/al/m a/pthanthe behavior as2as a fumtion of m ; we use -mratherto display results for values of m close to zero, sinceis obvious.m -)coInsert Figures A and B about hereTo separate the curves, those forn 3 and 5A and forIt is seen that1/2 p/a 2.in Figure B.enjoys a healthy advantage over11Whenis much smaller thanpnoticeable effect, and whenX.lin.a-in the regionathebias ofXaandphas awill usually coincidethe value ofis the con-MSE/62This value is approached by the curves for bothm gets large, but the1.1.is much larger the chances of a negativeNote that for the estimatorX .stantaspare slight so that the values of bothwithare shown in Figuren 2 and 4aand11curve gets there faster.aInsert Table-2 about hereTable 2 indicates the behavior of the MSE functions for larger valuesofnIt is clear that foras large as 9, the MSE/an,much greater at m -0 than thewant to avoid usingsome values of mItforMSE /a4n 9ofa2-of pis sois anywhere that one wouldeven though it is better thanafor

smalln, but asthere are certain advantages toMSE/a2are outweighed by a very highare normal; for veryXiThis gives the general picture when theincreases thesein the neighborhood of m. 0 .next question is, do the small-sample advantages ofX.nThepersist when theIIare distributed other than normally?In the one -way components of variance problem, the usual unbiasedestimator- Xof2of the main effect variance component has the distribution22G1X1 - G2X2 , where X 12and X2are independent chi-square randomvariables with known Aegrees of freedom01andGf1andf2, respectively, andare positive numbers related to the unknown variance -com2ponents and which satisfy the inequality E(X) Alfa. - A2f2are four cases to consider:highfland lowlow ff2 , and high1and lowffl -and high2, lowf2 .degrees of freedot have-been chosen as 2 and 20.f1.Thereand highf2,The low and highOther even values areeasily treated using the formulas of the next section.Odd nuMbered valuesare much more difficult mathematically, but by continuity we expect thecharacteristics to be similar to theadjacent even values.Insert Figures C and D about hereFigures C and D show, respectively, somef2 20and forfl 20 1f22, 2145E/aThe curves forcurves forfi f2 2fl 2 ,havebeen examined and their appearance is similar to Figure C, with thecurve having less upsweep nearforf1in 0 .Some values were also calculated f2 20 , and they were reasonably close to the values for the

-7normal case (see Table 3).high thr, estimatoris considerably better than1-1flThe conclusion may be drEhrn that whenhigh values of m , but that whenfl;for moderatelyis low the advantage is so meagerthat it does not seem worth the risk near m ' 0 .Insert Table 3 about herecan be considered to be aXiTo summarize; in problems where theweighted difference of two independent chi-squares, we would recommendaoveris quite small, the degrees of freedom of the first chi-nifsquare is fairly large; and it is thought likely that m 1/2the mean of Xi,i.e., thatis at least half its standard deviation.In a one-Way variance components model where it is desired to combineestimates, frome (between) and 2aestimatesbindependent sources, of the common variance componentsnd2(within); this means we recommend that negativew2ofabsmall, the number of leVels 2v12,a /(var a )2bbthe rationbe replaced by zero before averaging providedisin the one-way classification is large, andkis at least 1/2.From [2, page 322, formula(5.7)], the latter condition in. the balanced case is equivalent to22bwala (12lc/2(kr - 1 )/(kr - k))2] /r,whereris the number of1observations per level (within one source) and/ (2k - 3)2 .simply calculated quantity which exceeds the above lower bound forprovided kexample,11, 5 And r 2 ,is2//ris preferable toawhen.If k 262,2ab aw/35and(andA more2, 2a /ab wr 10 , fornis small).

-8-DERIVATION OF FORMULAS FOR MEAN SQUARE ERROR4.We now indicate how the.values for the curves discussed in Section 2were calculated.eAs before, denote by g andof thethe mean and variance, respectively, X, and for any U denote Uiformulas-for the MSE of a X max(000) .-12 and of XI ,- since theit withAppaii v and variance TMSE of any estimator of gTThese-can be obtainedand a (F.4 /n .from expressions for the mean and variance ofWe wish to deriveis given byagjIf F is the c.d.f. of X , the meinivand varianceT2of gare given byv nT2E(4) v2fxdF(x)E(002) (4.1),fx2dvx)(4.2).0For the normal case, letfz (x - g)/a ; thencoxdF(i)Jr.(g az)d0(z)0.ir02 , ,x dF(x) f-giag2 2gaz a2 2,zdot)iz,1

-gwhere 02using integration by parts onand2nTThe integration is straightforward,is the standard normal c.d.f.22 a Nmz d0(z)v2 a(m0(m) gm))Hence2where (m) (20-2 exp(-m /2),m 0(111)11 - gm)) - tO(m)(20(m) - 1) 0(m) - ( (0)and N2 Finally the MSE-forgis2a (( m /n) 4* 5m2 )where5MSE form 10(4.3)),- 0(m))Sincehas a normal distribution, thea -may -be derived similarly; its -value is(a2/n)tx5.(4.14)The distribution of a weighted difference of two-independent chisquares with even degrees of freedom can-be represented-as.a finite mixtureof positive and negative chi-squares.This result hasibeen derivedAriously by Box [11, Mantel and FaSternack I6], and Jayachandran-and Barr[5].Using the notation of Section 3 with D-e f1/220-X- - 0 X2 21integers the density o2 Xfx) Eq(Pi.1E1p- E c (r- p q)g (x) 'say, wheresquare withr 02/02j,gi(x)q f2/2ascan be written as(1 01"-j gi(x)(p q - k - 1)k 1and1.13op q-kh (x)k(4 .5 )d (r .m a)h ( )k xk 'is the density of 01 times a chi-degrees of freedom andhk(x)is the density of(-02)

-10-2ktimes a chi-square withwhenxg.(x)Note that theare nonzero onlyhk(x)is positive and thexare nonzero only whendegrees of freedom.Since (4.1) and (4.2) involve integration over onlyis negative.positive values ofx , we need consider only the terms containingThe expectation ofXI20 Xg.(x) .is therefore the weighted sum of expectations ofrandom variables with the weights given by the coe2ficients of the1 2jg.(x)in(4.5),andthesecondniomentofLis a similar weighted sum ofG X21 jsecond moments of thesea2 4.02(p r4q) ,.Using the fact that.pE c.(r;p,q)(2j01)E X.j 1 d(p r2-I P2 EiCj tr;15,0j 1andpE((X(!-)2) ;P,q)1)cl'20][a2/(1)j.12These formulas were used to compute MSE/a2a function ofr.The quantity MSE/afor the estimator[7.aswas then plotted as a function -ofm by obtaining m from the monotonically decreasing relation m(p - rq)(p2,2401(p r q)also2r q) 2r 0.which follows fromSinceu 0 , the maximum'value ofimplies that m p2 (f1/2)203 Gain ,04 02/n , andin the same manner as forILa2 2X3andp/q ;allowed is.2X42degrees of freedom, respectively.randhas the representationNote that the distribution ofwhereµ 201(p - rq)Thus MSE/ahavenfi(:)3Xand- A4X124ofmay be calculated forwith the appropriate substitutions.,

OTHER POSSIBLE ESTIMATORS5.In this section we look at a few other naturally suggested estimatorsMotivated by the resultSjust stated, the author decidedof simple form.to seek a simply-calculated-compromise betweenbetter thana41andti, one that doesfor larger valuet of m gia but does not have thecharacteristically high MSE thatghas for values of m near zero.Itis appealing to try to use the sample informatinn to get some idea of mand then chooseaorgbased on the proportion ofaccordingly.An easy rule to use would beXi which are negative.If this proportiongia , and weis high, it indicates a greater likelihood of low values ofwould perhaps be predispoSed to use the rule that performs best for theselow values.For breVity we restrict ourselves to the modelboth Xand X122haVe just two degrees of freedom.be the easiest case mathematically.fx(x) (01g1(x)OX.02X: whereThis turns out toFrom (4.5),02h1(x))/(01 02)2HereX.is distributed as a mixture of a- 0 Xand a1-022Xrandomvariable,each with 2 degrees of freedom, with weights proportional to01and02respectively.Regard the sampling as coming from an urnwith "positive" and "negative" balls in this proportion.sample it is easy to write the probabilities of observingballs ( Jtaking values from 0 toFor a smallJnegativen ) and to compute the *mean square

-12-Letting Yoraerror for any rule which choosesaccording to the value of J -ETstand for the resulting estimator,.MSE E(YPr(J j)E((Y - 41)21J j)g)2 n 3 , by symmetry we can writeTaking as an exampleE((Y - 021J 1) E( -(Y - g)21X1 0, x2 0, x3 0)When X1 0,so that here X.we haveX3 . 0X2 ). 0 , andµ (X3Xi 0 ,X2 X2.and4- x3)/3 , where X2 and X3 have dom variables, eachHencewith 2 degrees of freedom.289 /9The formula forditional distribution-of(02/3)Xb , where2XaE(µIIJ 1) 401/302IJE(671. 1)-2and XbThe con-follows directly.J 1, given that5Cand Var(lIJ 1) (01/3)Xis that ofare independent chi-squares withdegrees of freedom, respectively.-14. and 2May be calculated along similar lines to those indicated in Section 4.J 2J 0case is similar, and theIf Y2or X22aThus the first and second moments ofandJ 3Thecases are trivial.have an even number of degrees of freedom mnre thantwo, the same general principles may be used, but the distilbutions, givenwhether positive or negative, are no longer pure chi-squares but mixturesof chi-squares.Whenn 3The computations for athen become rather involved.there are just 4 possible rules of the type describedabove, based on choice ofawhich are simply a and µ .orµ(WhenwhenJ 1J 0 or 3 ,and when J 2 , two ofaandµyield the

-13-Let gsame value.)be the estimator obtained by choosing g whenI(This is the rule based on the heuristicJ 1 and a when J 2 .likelihood argument of the first paragraph.)obtained by choosing1.7.and awhen J 2Letbe the estimator172when J 1 .Insert Table 4 about hereFor selected values ofr 02/91 , Table 4 gives the correspondingfor the rules µ ,value of m via and MBE/62fl f2 2 .in the caseSurprisingly, the ruleIli,1.7.1and1:2/awhich choosesor- p according to the relative likelihood of low or high values of mwhich reverses the choice does relativelyperforms badly, and the rule g2well.ahas a lower MSE than112m 0 , where1.7.loses toworse than a by .031.a than is1.7.1.7.2oraand a lower MSE than eitherfor all m greater than about .28µfor m between .28 and .55.-a by .083, the compromise estimatorAt'12is thus a much more acceptable alternative to.We present a possible explanation for the fact that the MSE foris usually lower than forillprobably in underestimate of g .it , and hence that XThe estimator a ifestimatorbetter.is close to zero.pButfor largerg (i.e.,iscompensates forthis to some extent by frequently yielding zero in these cases.g17,2When two out of three of the xi are.negative, we know that these Xi are less thangood ifisThis isp. .28a ), thewhich is strictly positive in these cases seems to perform

When only oneXiis negative, on the other hand,be positive and as such will usually be closer tog .This holds unlessgR.will usuallythan will the biasedis quite high ( .55), where now even onegiais unlikely and when it occurs it is evidence thatnegative XiX willg , so again the estimate should be raised.underestimateIt should be pointed out that this article is written. from theirequentist point of view, according to which expectations are based onrepeated sampling with the same parameter values.This viewpoint hasbeen challenged by many statisticians (see, e.g., [7]), and alternativecriteria of performance might conceivably yield different results.Also included in Table it are results forthe sample median.Its mean and variance are easily calculated in the-f12 2a , the positive part ofcase since here21 1is exponentially distributed, andusing the no-memory property the order statistics are directly expressedas convolutions of exponential random variables; see, e.g., [3, P- 55,rrup. 3].performs better than its competitors when -m[aworse when -mis large.In the case of largeis small andm- the distribution ofXl is markedly skewed, and we would thus expect the sample median to becentered around the population median, but not the population mean.in -0 the distribution ofcasef1f2X.Asapproaches the double exponential, for the 2 , for which the sample median is the maximum-likelihoodestimator of the center of symmetry and is known to have good properties.This feature would not be expected to be present throughout all the distributions forX.studied here.

-15-CONCLUSION6.We have compared two simple rules for the problem of combiningindependent, identically distributed, unbiased estimates of a parametervalue known to be positive when the estimates may be negative.ruleThe first(the positive part of the average of the estimates) is consistent:1and the second rule11(the average of:the positive parts) is inconsistent.In large samples there is of course no difficulty in choosing between theWe have seer that whentwo.depends both on the ratiobetter; otherwiseais very small the relative performanceng/a(wheng/a 1/2,t71.is-generallyis better) and on the underlying distribution.rule of thumb for when to useIIAin practice is suggested in the last twoparagraphs of Section 3; however, this requires a prior idea of g/a andof the underlying family of distributions.A good practical rule is notobvious for situations where no such prior knowledge exists.For the case of the chi-square difference with (222)- degrees of free-dom andn 3 , the positive part of the sample median performs veryis largeiThe estimatorwell when g/ais small, but badly when g/ag2if two of the independent unbiased estimates are negative)(choosegperforms better than the others in the intermediate region and alsoreasonably mell for largeg/a.All estimators studied here have a positive probability of beingzero.It would be desirable to have a simply calculated estimator withthe property of always being (strictly) positive, or at least of beingpositive whenever one or more of theXiis positive.To date noreasonable method of obtaining such an estimator seems to be available.

-16-FOOTNOTES*T. W. F. Stroud is assistant professor, Department of Mathematics,queen's University, Kingston, Ontario, Canada.This research was performedwhile the author was visiting research fellow, Psychometric Group, EducationalTesting Service, Princeton, New Jersey.The author is grateful for thecomments of Donald B. Rubin and Robert I. Jennrich.The author wishesto thank Michael-W. Browne for suggesting the estimatorµ , and FredericM. Lord for some of the references.1For the normal case M co 1 and for the weighted difference of twochi-squares M is the square root of half the degrees of freedom of thepositive chi-sauare (see Section 4, second last paragraph).

-17-REFERENCES[1]Box, G. E. P., "Some'Theorems on Quadratic Forms Applied in theStudy of Analysis of Variance Problems, I.Effect of Inequality ofVariance in the One-Way Classification," Annals of MathematicalStatics 25. (June 195h), 290-302.[2]Brownlee; K. A., Statistical Theory and Methodology in Scienceand Engineering, 2nd ed., New York:[5]John Wiley & Sons, Inc., 1965.Chernoff, H., Gastwirth, J. L. and Johns, M.-V., Jr., "AsymptoticDistribution of Linear Combinations of Functions of Order Statisticswith Applications to Estimation," Annals of Mathematical Statistics,28, (February 1967), 52-72.[4]Henderson, C. R., "Estimation of Variance and.Covariance Components,"Biometrics, 2 (June 1953), 226-52.[5]Jayachandran, Tokeand Barr, D. R., "On the Distribution of aDifference of Two Scaled Chi-Square Random Variables,"Statistician[6]MantelAmerican2h (December 1970), 29-30.Nathan and Pasternack, Bernard S., "Light Bulb Statistics,"Journal of the American Statistical Association, 61 (September 1966),633-39.[7]Pierce, Donald A., "On Some Difficulties in a Frequency Theory ofInference," Annals of Statistics, 1 (March 1975), 241-50.[8]Rao, C. Radhakrishna, "Estimation of Variance and Covariance Componentsin Linear Models," Journal of the American Statistical Association,67 (March 1972), 112-15.

-18[9]Searle, Shayle R., Linear Models] New York:John Wiley & Sons, Inc.,1971.[10]Sirotnik, Ken, "An Analysis of Variance Framework for MatrixSampling," Educational and Psychological Measurement, 30 (Winter1970), 891-908.[11]Stroud, Thomas W. F., "Forecasting a Regression Relationship WhichVaries over a Large NuMber of Subpopulations," Research Bulletin73-00, Princeton, 11:J.:Educational Testing Service, 1973.

-19-1.SUMMARY OF PERFORMANCE OF2,31 2a12estimatorwith logermaximum11 betterworse114,5µRELATIVE TOa6-811-better12 bettervery closep. worse1.1.- 'worSep. mud, worsedepends ondistributionaaof X.1auMaximum" refers to max imum MSE/62 over any region K M , where- M is the largestof the form 0 .possible-value of

*94 Ta911o64Lo-aggoT6TT9ggoiuoLa* Udto NOSIHWT1400-aLitio*etagN92,91UOLEriVriUaINV6 rri?I) tSVO rIVEHOM)ri riTi(Tkih

a.083.136.166.167a.167a.167a00.40.81.21.62.02 f 2 .Values have been extrapolated mathematically;m not exist 1 f2 .250a 2.167af1 f2COMPARISON OF NORMAL CASE WITH CHI-SQUARE DIbFbRENCE ( a2 1 )03 N1-

-22-VALUES OF MSE/a4.2FORa ,n 3 AND111o2ANDaWHENLI f2 2MSE/02r 1.00.167.250.219.198.160o0.2.

.3.4IMSE /Q2.2II.41A.or.OOPi?ISE comparison forONO ONO(na4)41101.11111.0.011111111.(normal case).1.41.6.4111111111M1.84. .1112.04111 MONIO1 111111111n 2 and 4pia111111TE(n 2)

.2.3.5-IMSE/c2IC.2i4Ii.6iiI.8In 5)II1.0( fl m 2F. (n1 2)MSE comparison for chi-square differenceIA (nx2)i,If2 . 20 ).1.2I11.4

positive parts before averaging. Denote,(R) , the estimator obtained by the first method, by a, since it is the maximum-likelihood estimator under normality. We denote the second estimator (E4)/n by µ , since it is the arithmetic mean of a set of quantities. These two estimators were con-

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SR333HD Max-E-Therm 333 BTU Heavy Duty Heater 333 138 SR400NA Max-E-Therm 400 BTU Natural Gas Heater 400 138 SR400LP Max-E-Therm 400 BTU Propane Gas Heater 400 138 SR400HD Max-E-Therm 400 BTU Heavy Duty Heater 400 138 460763 Max-E-Therm 400 BTU ASME NA 400 149 460764 Max-E-Therm 400 BTU ASME LP 400

SR333HD Max-E-Therm 333 BTU Heavy Duty Heater 333 138 SR400NA Max-E-Therm 400 BTU Natural Gas Heater 400 138 SR400LP Max-E-Therm 400 BTU Propane Gas Heater 400 138 SR400HD Max-E-Therm 400 BTU Heavy Duty Heater 400 138 460763 Max-E-Therm 400 BTU ASME NA 400 149 460764 Max-E-Therm 400 BTU ASME LP 400

1 Amp Standard SCRs Product Selector Part Number Voltage Gate Sensitivity Type Package 400V 600V 800V 1000V Sx01E X X 10mA Standard SCR TO-92 SxN1 X X 10mA Standard SCR Compak Note: x Voltage Dimensions - Compak (C Package) 0.079 (2.0) 0.040 (1.0) 0.030 (0.76) 0.079 (2.0) 0.079 (2.0) 0.110 (2.8) Pad Outl

H 079 370 80 29 h.bertschinger@bluewin.ch T 044 784 91 55 H 079 406 84 49 kolleroptik@bluewin.ch T 052 346 15 47 meibau@bluewin.ch H 079 413 28 30 ZKAV-Rechnungsprüfungs-Kommission (RPK) Bertschinger Hansruedi Brunnenstr. 10 8632 Tann Hiestand Robert Bruggacker 9 8488 Turbenthal Röthlin Erich Sunnhaldestr. 23 8489 Wildberg Sigg Rudolf .